The figure below shows the ellipse $\frac{(x-20)^2}{20}+\frac{(y-16)^2}{16}=2016$.

[asy]
defaultpen(linewidth(0.7));
pair c=(20,16);
real dist = 30;
real a = sqrt(2016*20),b=sqrt(2016*16);
xaxis("x",c.x-a-dist,c.x+a+3*dist,EndArrow);
yaxis("y",c.y-b-dist*2,c.y+b+3*dist,EndArrow);
draw(ellipse(c,a,b));
label("$R_1$",(100,100));
label("$R_2$",(-80,100));
label("$R_3$",(-60,-75));
label("$R_4$",(70,-75));
[/asy]
Let $R_1,R_2,R_3,$ and $R_4$ denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of $R_1-R_2+R_3-R_4$.

1280

To find the value of $R_1-R_2+R_3-R_4$, we need to find the areas of $R_1$, $R_2$, $R_3$, and $R_4$ individually.

To find the area of each region, we need to understand the geometry of the ellipse.

The equation of the ellipse given is $\frac{(x-20)^2}{20}+\frac{(y-16)^2}{16}=2016$.
The centers of the ellipse are at $(20, 16)$, and the lengths of the semi-major and semi-minor axes are $\sqrt{2016 \cdot 20}$ and $\sqrt{2016 \cdot 16}$, respectively.

Now, let's find the areas of each region:

1. $R_1$ (first quadrant):
To find the area of $R_1$, we need to find the portion of the ellipse that lies in the first quadrant. This region is bounded by the x-axis, the y-axis, and the ellipse itself.

Since the ellipse is symmetric with respect to both the x-axis and the y-axis, we can calculate the area of $R_1$ as follows:
$$R_1 = \text{Area of the ellipse}/4$$

2. $R_2$ (second quadrant):
To find the area of $R_2$, we need to find the portion of the ellipse that lies in the second quadrant. This region is symmetric to $R_1$ with respect to the y-axis.

Similarly, we can calculate the area of $R_2$ as follows:
$$R_2 = \text{Area of the ellipse}/4$$

3. $R_3$ (third quadrant):
To find the area of $R_3$, we need to find the portion of the ellipse that lies in the third quadrant. This region is symmetric to $R_1$ with respect to both the x-axis and the y-axis.

Again, we can calculate the area of $R_3$ as follows:
$$R_3 = \text{Area of the ellipse}/4$$

4. $R_4$ (fourth quadrant):
To find the area of $R_4$, we need to find the portion of the ellipse that lies in the fourth quadrant. This region is symmetric to $R_1$ with respect to the x-axis.

We can calculate the area of $R_4$ as follows:
$$R_4 = \text{Area of the ellipse}/4$$

Now, substitute the area of the ellipse into the equations for $R_1$, $R_2$, $R_3$, and $R_4$ to find their respective areas.

Finally, subtract $R_2$ from $R_1$ and $R_4$ from $R_3$ to get $R_1-R_2+R_3-R_4$.